An epitrochoid is defined as a roulette formed when a first circle rolls around the outside of a second circle. The first circle is called the rolling generating circle. The second circle is called the fixed generating circle. The trochoid is called a limaçon when the diameter of the fixed circle and the rolling generating circle are equal. The equation of a limas on in polar coordinates has the form r=b+a cos α. The epitrochoid is called a Wankel type when the diameter of the fixed circle is twice that of the rolling generating circle. (The cylinder of the Wankel engine is an epitrochoid.)
When b>a, the limaçon is a single-loop limaçon and has no inner loop, and the rotating piston has two sharp corners. Pistons with sharp corners have problems with sealings and leaks. There are hundreds of patents disclosing systems in which b>a. Early examples include Woodhouse's rotary steam engine from 1839 and U.S. Pat. No. 298,952 from 1884, and recent examples include U.S. Pat. No. 8,539,931 and EP Patent Publication No. 0 310 549 (see, e.g., FIG. 1 of the present application). A fixed single loop limaçon cylinder with an orbiting piston has been in the public domain for more than 175 years.
FIG. 1 shows a conventional fixed single-loop limaçon cylinder 106 and a piston 105 with sharp corners. The piston 105 rotates around an orbital axis 101, and the orbital axis 101 moves circularly around a fixed axis 102 that is parallel to the orbital axis. 103 is an intake port. 104 is an exhaust port. 108 is a compression space, and 107 is an intake space.
If b<a, the limaçon is a dual-loop limaçon and has an external loop and an internal loop. The piston has the form of an ellipse with a major axis equal to a+b and a minor axis equal to a-b. Examples of a fixed limaçon external loop cylinder with an orbital elliptic piston include U.S. Pat. Nos. 3,387,772 and 6,926,505 and US Patent Application Publication No. 2011/0200476.
FIG. 2 shows a cross section of a conventional fixed limaçon cylinder 114 and an elliptic piston 113. The cylinder 114 has a shape that corresponds to the external loop of a dual-loop limaçon. The piston 113 rotates around an orbital axis 112, and the orbital axis 112 moves circularly around a fixed axis 111 that is parallel to the orbital axis 112. 115 is an exhaust port. 116 is a compression space, and 117 is an intake space.
A piston rotating inside a fixed cylinder with limaçon cross-section will always have at least two lines of contact with the cylinder wall. The piston rotates around a first axis, and the first axis simultaneously makes a circular orbital motion around another axis that is fixed relative to that limaçon cylinder and that is parallel to the first axis. The ratio between the rotation of the piston around the center of the piston and the circular motion of the first axis around the center of the circular motion is 1:2 (see, e.g., the example of FIG. 3). (In the Wankel engine, the corresponding relation between the rotation of the piston and the orbital angular motion is 3:2.)
A piston with an internal loop limaçon cross-section rotating inside a fixed elliptic cylinder always has at least two lines of contact. The piston rotates one turn counterclockwise when the axis of rotation makes one turn clockwise (e.g., in the opposite direction).
In an Otto or Diesel engine, 29% of the energy in the fuel is transferred to the cooling system, and 33% goes to the exhaust system. With hot cylinder walls, the cooling can virtually disappear. With a higher expansion ratio than compression ratio, the exhaust losses can diminish. Losses due to friction between the piston and the cylinder are also diminished.
An n-step, n+1 volume, volume-to-volume expander uses a relatively small first displacement space. The first displacement gas space is connected to a high pressure gas source and filled with an amount (mass) of gas. The amount of gas is transferred to a bigger second displacement space. The transfer of the amount of gas from a smaller to a bigger displacement space is repeated n times in a cycle. The (n+1)th (or last) displacement space is connected to a low pressure gas sink and emptied with the working gas.
An n-step, volume-to-volume expander needs n+1 expansion volumes in order to do n expansion steps. Shanghai Jiaotong University (report to the International Compressor Engineering Conference at Purdue Univ., July 2010) and Daikin (U.S. Pat. No. 7,896,627) disclose volume-to-volume expanders using the principle in their experimental rolling piston expanders. U.S. Pat. No. 6,877,314 and U.S. Pat. No. 8,220,381 disclose free piston, one-step, volume-to-volume expanders. U.S. Pat. No. 8,695,335 discloses a liquid ring volume-to-volume expander.
This “Discussion of the Background” section is provided for background information only. The statements in this “Discussion of the Background” are not an admission that the subject matter disclosed in this “Discussion of the Background” section constitutes prior art to the present disclosure, and no part of this “Discussion of the Background” section may be used as an admission that any part of this application, including this “Discussion of the Background” section, constitutes prior art to the present disclosure.